Annotation of rpl/lapack/lapack/dlatrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
! 2: *
! 3: * -- LAPACK auxiliary routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: CHARACTER UPLO
! 10: INTEGER LDA, LDW, N, NB
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
! 14: * ..
! 15: *
! 16: * Purpose
! 17: * =======
! 18: *
! 19: * DLATRD reduces NB rows and columns of a real symmetric matrix A to
! 20: * symmetric tridiagonal form by an orthogonal similarity
! 21: * transformation Q' * A * Q, and returns the matrices V and W which are
! 22: * needed to apply the transformation to the unreduced part of A.
! 23: *
! 24: * If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
! 25: * matrix, of which the upper triangle is supplied;
! 26: * if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
! 27: * matrix, of which the lower triangle is supplied.
! 28: *
! 29: * This is an auxiliary routine called by DSYTRD.
! 30: *
! 31: * Arguments
! 32: * =========
! 33: *
! 34: * UPLO (input) CHARACTER*1
! 35: * Specifies whether the upper or lower triangular part of the
! 36: * symmetric matrix A is stored:
! 37: * = 'U': Upper triangular
! 38: * = 'L': Lower triangular
! 39: *
! 40: * N (input) INTEGER
! 41: * The order of the matrix A.
! 42: *
! 43: * NB (input) INTEGER
! 44: * The number of rows and columns to be reduced.
! 45: *
! 46: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
! 47: * On entry, the symmetric matrix A. If UPLO = 'U', the leading
! 48: * n-by-n upper triangular part of A contains the upper
! 49: * triangular part of the matrix A, and the strictly lower
! 50: * triangular part of A is not referenced. If UPLO = 'L', the
! 51: * leading n-by-n lower triangular part of A contains the lower
! 52: * triangular part of the matrix A, and the strictly upper
! 53: * triangular part of A is not referenced.
! 54: * On exit:
! 55: * if UPLO = 'U', the last NB columns have been reduced to
! 56: * tridiagonal form, with the diagonal elements overwriting
! 57: * the diagonal elements of A; the elements above the diagonal
! 58: * with the array TAU, represent the orthogonal matrix Q as a
! 59: * product of elementary reflectors;
! 60: * if UPLO = 'L', the first NB columns have been reduced to
! 61: * tridiagonal form, with the diagonal elements overwriting
! 62: * the diagonal elements of A; the elements below the diagonal
! 63: * with the array TAU, represent the orthogonal matrix Q as a
! 64: * product of elementary reflectors.
! 65: * See Further Details.
! 66: *
! 67: * LDA (input) INTEGER
! 68: * The leading dimension of the array A. LDA >= (1,N).
! 69: *
! 70: * E (output) DOUBLE PRECISION array, dimension (N-1)
! 71: * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
! 72: * elements of the last NB columns of the reduced matrix;
! 73: * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
! 74: * the first NB columns of the reduced matrix.
! 75: *
! 76: * TAU (output) DOUBLE PRECISION array, dimension (N-1)
! 77: * The scalar factors of the elementary reflectors, stored in
! 78: * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
! 79: * See Further Details.
! 80: *
! 81: * W (output) DOUBLE PRECISION array, dimension (LDW,NB)
! 82: * The n-by-nb matrix W required to update the unreduced part
! 83: * of A.
! 84: *
! 85: * LDW (input) INTEGER
! 86: * The leading dimension of the array W. LDW >= max(1,N).
! 87: *
! 88: * Further Details
! 89: * ===============
! 90: *
! 91: * If UPLO = 'U', the matrix Q is represented as a product of elementary
! 92: * reflectors
! 93: *
! 94: * Q = H(n) H(n-1) . . . H(n-nb+1).
! 95: *
! 96: * Each H(i) has the form
! 97: *
! 98: * H(i) = I - tau * v * v'
! 99: *
! 100: * where tau is a real scalar, and v is a real vector with
! 101: * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
! 102: * and tau in TAU(i-1).
! 103: *
! 104: * If UPLO = 'L', the matrix Q is represented as a product of elementary
! 105: * reflectors
! 106: *
! 107: * Q = H(1) H(2) . . . H(nb).
! 108: *
! 109: * Each H(i) has the form
! 110: *
! 111: * H(i) = I - tau * v * v'
! 112: *
! 113: * where tau is a real scalar, and v is a real vector with
! 114: * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
! 115: * and tau in TAU(i).
! 116: *
! 117: * The elements of the vectors v together form the n-by-nb matrix V
! 118: * which is needed, with W, to apply the transformation to the unreduced
! 119: * part of the matrix, using a symmetric rank-2k update of the form:
! 120: * A := A - V*W' - W*V'.
! 121: *
! 122: * The contents of A on exit are illustrated by the following examples
! 123: * with n = 5 and nb = 2:
! 124: *
! 125: * if UPLO = 'U': if UPLO = 'L':
! 126: *
! 127: * ( a a a v4 v5 ) ( d )
! 128: * ( a a v4 v5 ) ( 1 d )
! 129: * ( a 1 v5 ) ( v1 1 a )
! 130: * ( d 1 ) ( v1 v2 a a )
! 131: * ( d ) ( v1 v2 a a a )
! 132: *
! 133: * where d denotes a diagonal element of the reduced matrix, a denotes
! 134: * an element of the original matrix that is unchanged, and vi denotes
! 135: * an element of the vector defining H(i).
! 136: *
! 137: * =====================================================================
! 138: *
! 139: * .. Parameters ..
! 140: DOUBLE PRECISION ZERO, ONE, HALF
! 141: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
! 142: * ..
! 143: * .. Local Scalars ..
! 144: INTEGER I, IW
! 145: DOUBLE PRECISION ALPHA
! 146: * ..
! 147: * .. External Subroutines ..
! 148: EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
! 149: * ..
! 150: * .. External Functions ..
! 151: LOGICAL LSAME
! 152: DOUBLE PRECISION DDOT
! 153: EXTERNAL LSAME, DDOT
! 154: * ..
! 155: * .. Intrinsic Functions ..
! 156: INTRINSIC MIN
! 157: * ..
! 158: * .. Executable Statements ..
! 159: *
! 160: * Quick return if possible
! 161: *
! 162: IF( N.LE.0 )
! 163: $ RETURN
! 164: *
! 165: IF( LSAME( UPLO, 'U' ) ) THEN
! 166: *
! 167: * Reduce last NB columns of upper triangle
! 168: *
! 169: DO 10 I = N, N - NB + 1, -1
! 170: IW = I - N + NB
! 171: IF( I.LT.N ) THEN
! 172: *
! 173: * Update A(1:i,i)
! 174: *
! 175: CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
! 176: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
! 177: CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
! 178: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
! 179: END IF
! 180: IF( I.GT.1 ) THEN
! 181: *
! 182: * Generate elementary reflector H(i) to annihilate
! 183: * A(1:i-2,i)
! 184: *
! 185: CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
! 186: E( I-1 ) = A( I-1, I )
! 187: A( I-1, I ) = ONE
! 188: *
! 189: * Compute W(1:i-1,i)
! 190: *
! 191: CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
! 192: $ ZERO, W( 1, IW ), 1 )
! 193: IF( I.LT.N ) THEN
! 194: CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
! 195: $ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
! 196: CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
! 197: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
! 198: $ W( 1, IW ), 1 )
! 199: CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
! 200: $ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
! 201: CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
! 202: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
! 203: $ W( 1, IW ), 1 )
! 204: END IF
! 205: CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
! 206: ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
! 207: $ A( 1, I ), 1 )
! 208: CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
! 209: END IF
! 210: *
! 211: 10 CONTINUE
! 212: ELSE
! 213: *
! 214: * Reduce first NB columns of lower triangle
! 215: *
! 216: DO 20 I = 1, NB
! 217: *
! 218: * Update A(i:n,i)
! 219: *
! 220: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
! 221: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
! 222: CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
! 223: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
! 224: IF( I.LT.N ) THEN
! 225: *
! 226: * Generate elementary reflector H(i) to annihilate
! 227: * A(i+2:n,i)
! 228: *
! 229: CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
! 230: $ TAU( I ) )
! 231: E( I ) = A( I+1, I )
! 232: A( I+1, I ) = ONE
! 233: *
! 234: * Compute W(i+1:n,i)
! 235: *
! 236: CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
! 237: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
! 238: CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
! 239: $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
! 240: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
! 241: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
! 242: CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
! 243: $ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
! 244: CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
! 245: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
! 246: CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
! 247: ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
! 248: $ A( I+1, I ), 1 )
! 249: CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
! 250: END IF
! 251: *
! 252: 20 CONTINUE
! 253: END IF
! 254: *
! 255: RETURN
! 256: *
! 257: * End of DLATRD
! 258: *
! 259: END
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